Implications of a Froissart bound saturation of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mi>γ</mml:mi><mml:mo>*</mml:mo></mml:msup><mml:mtext mathvariant="normal">−</mml:mtext><mml:mi>p</mml:mi></mml:math>deep inelastic scattering. I. Quark distributions at ultra small<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>x</mml:mi></mml:math>

We argue that the deep inelastic structure function F_2^{\gamma p}(x, Q^2) regarded as a cross section for virtual \gamma^*pscattering, is hadronic in nature. This implies that its growth is limited by the Froissart bound at high hadronic energies, giving a \ln^2 (1/x)bound on F_2^{\gamma p}as Bjorken x\rightarrow 0 The same bound holds for the individual quark distributions. In earlier work, we obtained a very accurate global fit...